# Two persons A and B are throwing an unbiased six faced die alternatively, with the condition that the person who throws $3$ first wins the game. If A starts the game, the probabilities of A and B to win the game are respectively

$\left( a \right)\dfrac{6}{{11}},\dfrac{5}{{11}}$

$\left( b \right)\dfrac{5}{{11}},\dfrac{6}{{11}}$

$\left( c \right)\dfrac{8}{{11}},\dfrac{3}{{11}}$

$\left( d \right)\dfrac{3}{{11}},\dfrac{8}{{11}}$

Answer

Verified

366.9k+ views

Hint: Use formula of sum of infinite geometric series $S = \dfrac{a}{{1 - r}}$

Winning the game is getting a $3$ on the die $p\left( {getting 3} \right) = \dfrac{1}{6}$ and $q\left( {not getting 3} \right) = \dfrac{5}{6}$

A wins if he gets $3$ on his first turn or he gets $3$ on his second turn but B does not get $3$ on his first turn and so on.

$

p\left( A \right) = p + p{q^2} + p{q^4} + ....... \\

\Rightarrow p\left( {1 + {q^2} + {q^4} + ........} \right) \\

$

We can see sum of infinite geometric series formed of common ratio $r = {q^2}$and first term $a = 1$ then sum of infinite geometric series is $S = \dfrac{a}{{1 - r}}$

$\left( {1 + {q^2} + {q^4} + ........} \right) = \dfrac{1}{{1 - {q^2}}}$

$

\Rightarrow p\left( {1 + {q^2} + {q^4} + ........} \right) \\

\Rightarrow p\left( {\dfrac{1}{{1 - {q^2}}}} \right) \\

\Rightarrow \dfrac{p}{{1 - {q^2}}} \\

\Rightarrow \dfrac{{\dfrac{1}{6}}}{{1 - {{\left( {\dfrac{5}{6}} \right)}^2}}} \Rightarrow \dfrac{6}{{11}} \\

p\left( A \right) = \dfrac{6}{{11}} \\

p\left( B \right) = 1 - p\left( A \right) = 1 - \dfrac{6}{{11}} \\

p\left( B \right) = \dfrac{5}{{11}} \\

$

Probability of A to win the game is $p\left( A \right) = \dfrac{6}{{11}}$ .

Probability of B to win the game is $p\left( B \right) = \dfrac{5}{{11}}$.

So, the correct option is (A).

Note: Whenever we come across these types of problems first find the probability of winning or losing the game in the first attempt but we know it is not possible to win the game in the first attempt. So, we try unless anyone is not winning the game then we use a sum of infinite geometric series.

Winning the game is getting a $3$ on the die $p\left( {getting 3} \right) = \dfrac{1}{6}$ and $q\left( {not getting 3} \right) = \dfrac{5}{6}$

A wins if he gets $3$ on his first turn or he gets $3$ on his second turn but B does not get $3$ on his first turn and so on.

$

p\left( A \right) = p + p{q^2} + p{q^4} + ....... \\

\Rightarrow p\left( {1 + {q^2} + {q^4} + ........} \right) \\

$

We can see sum of infinite geometric series formed of common ratio $r = {q^2}$and first term $a = 1$ then sum of infinite geometric series is $S = \dfrac{a}{{1 - r}}$

$\left( {1 + {q^2} + {q^4} + ........} \right) = \dfrac{1}{{1 - {q^2}}}$

$

\Rightarrow p\left( {1 + {q^2} + {q^4} + ........} \right) \\

\Rightarrow p\left( {\dfrac{1}{{1 - {q^2}}}} \right) \\

\Rightarrow \dfrac{p}{{1 - {q^2}}} \\

\Rightarrow \dfrac{{\dfrac{1}{6}}}{{1 - {{\left( {\dfrac{5}{6}} \right)}^2}}} \Rightarrow \dfrac{6}{{11}} \\

p\left( A \right) = \dfrac{6}{{11}} \\

p\left( B \right) = 1 - p\left( A \right) = 1 - \dfrac{6}{{11}} \\

p\left( B \right) = \dfrac{5}{{11}} \\

$

Probability of A to win the game is $p\left( A \right) = \dfrac{6}{{11}}$ .

Probability of B to win the game is $p\left( B \right) = \dfrac{5}{{11}}$.

So, the correct option is (A).

Note: Whenever we come across these types of problems first find the probability of winning or losing the game in the first attempt but we know it is not possible to win the game in the first attempt. So, we try unless anyone is not winning the game then we use a sum of infinite geometric series.

Last updated date: 27th Sep 2023

•

Total views: 366.9k

•

Views today: 5.66k

Recently Updated Pages

What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Is current density a scalar or a vector quantity class 12 physics JEE_Main

An alternating current can be produced by A a transformer class 12 physics CBSE

What is the value of 01+23+45+67++1617+1819+20 class 11 maths CBSE

Which are the Top 10 Largest Countries of the World?

How many millions make a billion class 6 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Number of Prime between 1 to 100 is class 6 maths CBSE